Abstract: Motivated in part by questions from Number Theory, we consider trigonometric polynomials P_N with (completely) multiplicative random signs. I will talk about the distribution of P_N at a typical point on the unit circle and the size of the maximum of |P_N| on the circle, as N tends to infinity.

Our results correspond to classical results of Salem and Zygmund (where the coefficients are independent). However, in contrast to the case of i.i.d. coefficients, high moments of P_N are no longer asymptotically Gaussian. This makes the study of the maximum more challenging, and I will report on some partial results.

Based on a joint work with Jacques Benatar and Brad Rodgers.

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